MATH114 Integration and Differentiation

Exercises Week 4

[2.5ex]

Workshop questions that you cannot solve during the workshop should be regarded as additional training material and solved at home. Afterwards please compare with the model solutions. There are two marks for every correctly answered quiz question. Submission deadline for written assessment: Wednesday 16:00 in your tutor’s homework box. Submission deadline for quizzes: Wednesday 23:59 on Moodle.

Workshop

• True or false? Think for 1 minute, then resolve the quiz and discuss briefly.

  • The function $f:[0,1]\to ℝ$ defined by $f(x)=|x|$ is differentiable.

  • The function $g:[-1,1]\to ℝ$ defined by $g(x)=|x|$ is differentiable.

  • The function $h:ℝ\setminus \{0\}\to ℝ$ defined by $h(x)=|x|$ is differentiable.

• Let $\alpha >0$ be a rational number and

  $$f:[0,\mathrm{\infty })\to ℝ,f(x)=x^{\alpha }\sin \left(\frac{1}{x}\right).$$

  Determine for which $\alpha$ values the function $f$ is differentiable. (Hint: split into the two cases $x_0=0$ and $x_0\ne 0$ and use Example 3.2.8.)

• Prove the following statement:

  Let $f:I\to ℝ$ be strictly monotone and continuous and write $y_0:=f(x_0)$ and $y=f(x)$ if $x_0,x\in I$. Then

  $$y\to y_0\text{ if and only if }x\to x_0.$$

Written Assessment

• Consider the following statement with proof:

  Statement. If $I$ is an interval and a function $f:I\to ℝ$ is strictly increasing, bounded and continuous, then $I$ must be a bounded interval.
  Proof. We prove this by contradiction. Suppose $I$ were an unbounded interval, e.g. suppose $I$ were unbounded to the right. Then there would be an increasing sequence ${(x_n)}_{n\in \mathbb{N}}$ in $I$ tending to $\mathrm{\infty }$. Then ${(f(x_n))}_{n\in \mathbb{N}}$ would be increasing as well and tend to $\mathrm{\infty }$ because $f$ is increasing and ${(x_n)}_{n\in \mathbb{N}}$ tends to $\mathrm{\infty }$. But this means that $f$ would be unbounded, which contradicts the assumptions of the statement. A similar argument works if $I$ is unbounded to the left, in which case we would consider a decreasing sequence tending to $-\mathrm{\infty }$, leading to a similar contradiction. Thus $I$ must be bounded. $\mathrm{\square }$

  Decide whether the proof is correct or wrong. If correct, then explain and defend every single line of the proof. Instead if the proof contains mistakes, explain what exactly is wrong and prove that it is wrong, for example by finding a counter-example for the wrong bit. Decide whether the statement itself is true or false. If false, find a counter-example for the whole statement. [6]

• Prove Corollary 3.3.2(i): Suppose $f:[a,b]\to ℝ$ is a continuous function which is differentiable on $(a,b)$. If $f^{\prime }(x)\ge 0$, for all $x\in (a,b)$, then $f$ is increasing on $[a,b]$. [4]

Quiz

• Chain rule and Co. Consider the functions $f:(0,1)\to ℝ$ given by $f(x)=\arctan (x)$ and $g:(-1,0)\to ℝ$ given by $g(x)=x^2$. What can you do?

  • Get (only) ${(fg)}^{\prime }$ from the product rule.

  • Get (only) ${(\frac{f}{g})}^{\prime }$ from the quotient rule.

  • Get (only) ${(f\circ g)}^{\prime }$ from the chain rule.

  • Get (only) ${(f+g)}^{\prime }$ from the sum rule.

  • Get ${(fg)}^{\prime }$ from the product rule and ${(f+g)}^{\prime }$ from the sum rule.

• Mixed properties of functions. Consider a function $f:I\to ℝ$. Which of the following statements is true?

  • If $f$ is differentiable then it is continuous and constant.

  • If $f$ is continuous then it is differentiable.

  • If $f$ is constant and continuous then it is differentiable.

  • If $f$ is continuous then it is invertible.

  • If $f$ is strictly increasing and continuous then it is differentiable.

• Scrutinising a function. Consider the function $f:[0,\frac{\pi }{2}]\to ℝ$ defined by $f(x)=|\sin (x)|$. Which of the following statements is true?

  • $f$ is continuous, strictly increasing, differentiable with $f^{\prime }(x)=\cos (x)$, and invertible.

  • $f$ is continuous but not differentiable and not invertible.

  • $f$ is continuous and strictly increasing but not differentiable at $0$.

  • $f$ is continuous and increasing but not strictly increasing.

  • $f$ is discontinuous at $0$ but continuous and differentiable everywhere else.

• Differentiable functions. Which of the following functions are differentiable?

  • $a:ℝ\setminus \{0\}\to ℝ,a(x)=|x|$

  • $b:ℝ\setminus \{0\}\to ℝ,b(x)=|x+1|$

• Maximum of a function. What is the maximum of the function $f:ℝ\to ℝ$ defined by $f(x)=x{e}^{-x}$?

  • $2e^{â\mathrm{ˆ}\mathrm{’}2}$

  • $e^{-1}$

  • $1$

  • $\mathrm{\infty }$

  • None of the above